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**Benmath** Let V be an inner product space. Let $\displaystyle \{e_n\}_1^\infty$ be a complete orthonormal sequence in V. Suppose that $\displaystyle f,g \in V$ with $\displaystyle f = \sum_{n=1}^{\infty}c_ne_n$ and $\displaystyle g= \sum_{k=1}^\infty d_ke_k$. Prove that $\displaystyle \langle f, g \rangle = \sum_{n=1}^\infty c_n \bar{d_n}$.

My proof so far:

$\displaystyle \langle f, g \rangle = \langle \sum_{n=1}^{\infty}c_ne_n , \sum_{k=1}^\infty d_ke_k \rangle$

$\displaystyle = \sum_{n=1}^{\infty}c_n \langle e_n , \sum_{k=1}^\infty d_ke_k \rangle $

$\displaystyle = \sum_{n=1}^{\infty}c_n \bar{d_n} \langle e_n, e_n \rangle $

I definitely don't think I can pull that sum out in the first step because the $\displaystyle e_n$ has the sum with it as well. I was thinking of trying the following:

$\displaystyle \langle \lim_{n\to \infty} \sum_{n=1}^N c_n e_n , \lim_{k \to \infty} \sum_{k=1}^K d_k e_k \rangle$

I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.